#96393

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,20,23,21,24,28,31)(19,25,27,32,26,22,29,30)(33,34)(35,43)(36,42)(37,41)(38,40)(39,44)(45,47)(46,48);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,10,14,15)(17,19)(18,22,24,25)(20,27)(21,26)(23,30,31,32)(28,29)(33,35,37,45,38,46,36,44)(34,39,42,48,40,47,41,43);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(34,40)(39,47)(41,42)(43,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cgi,d^b=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3589);